Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equation. (English) Zbl 0743.35029
L’auteur écrit l’équation de Kuramoto-Sivashinsky sous la forme
\[
u'(t)+A(u)-A^{1/2}(u)+B(u,u)=0,\quad u(0)=u_ 0,
\]
avec \(u\in H=\{u=\sum^ \infty_{-\infty}iu_ ke^{i(2\pi kx/L)}\), \(u_ k=- u_{-k}\), \(\Sigma| u_ k|^ 2<\infty\}\). Il introduit des sous-espaces vectoriels de dimension finie de \(H\) pour obtenir une variété inertiable approchée (th. 4). Il montre au préalable une majoration de \(| A^ su(t)|\) (pour \(s\geq 0\)) pour \(t\) assez grand (th. 1), puis une propriété d’analyticité par rapport à la variable temps (th. 2).
Reviewer: M.-T.Lacroix (Saone)
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
References:
| [1] | Constantin, P.; Foias, C., Navier-Stokes Equations (1988), The University of Chicago Press: The University of Chicago Press Chicago · Zbl 0687.35071 |
| [2] | Constantin, P.; Foias, C.; Nicolaenko, B.; Temam, R., Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, (Applied Mathematics Sciences (1989), Springer: Springer Berlin), No. 70 · Zbl 0683.58002 |
| [3] | Constantin, P.; Foias, C.; Nicolaenko, B.; Temam, R., Spectral barriers and inertial manifolds for dissipative partial differential equations, J. Dynam. Diff. Eqs., 1, 45-73 (1989) · Zbl 0701.35024 |
| [4] | Foias, C.; Jolly, M. S.; Kevrekidis, I. G.; Sell, G. R.; Titi, E. S., On the computation of inertial manifolds, Phys. Lett. A, 131, 433-436 (1988) |
| [5] | Foias, F.; Manley, O.; Temam, R., Modelling of the interaction of small and large eddies in two dimensional turbulent flows, Math. Modelling Numerical Anal., 22, 93-118 (1988) · Zbl 0663.76054 |
| [6] | Foias, C.; Nicolaenko, B.; Sell, G. R.; Temam, R., Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimensions, J. Math. Pures Appl., 67, 197-226 (1988) · Zbl 0694.35028 |
| [7] | Foias, C.; Saut, J.-C., Remarques sur les equations de Navier-Stokes stationnaires, Ann. Scuola Norm. Sup. Pisa Ser., 4, 10, 169-177 (1983) · Zbl 0546.35058 |
| [8] | Foias, C.; Sell, G. R.; Titi, E. S., Exponential tracking and approximation of inertial manifolds for dissipative equations, J. Dynam. Diff. Eqs., 1, 199-224 (1989) · Zbl 0692.35053 |
| [9] | Foias, C.; Sell, G. R.; Temam, R., Inertial manifolds for nonlinear evolutionary equations, J. Diff. Eqs., 73, 309-353 (1988) · Zbl 0643.58004 |
| [10] | Foias, C.; Temam, R., Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Ann. Scuola Norm. Sup. Pisa Ser., 4, 5, 29-63 (1978) · Zbl 0384.35047 |
| [11] | Foias, C.; Temam, R., Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. Math. Pures Appl., 58, 339-368 (1979) · Zbl 0454.35073 |
| [12] | Foias, C.; Temam, R., Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87, 359-369 (1989) · Zbl 0702.35203 |
| [13] | F. Jauberteau, C. Rosier and R. Temam, The nonlinear Galerkin method in computational fluid dynamics, Appl. Num. Math., to appear.; F. Jauberteau, C. Rosier and R. Temam, The nonlinear Galerkin method in computational fluid dynamics, Appl. Num. Math., to appear. · Zbl 0702.76077 |
| [14] | Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S., Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations, Physica D, 44, 38-60 (1990) · Zbl 0704.58030 |
| [15] | Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S., Preserving dissipation in approximate inertial forms for the Kuramoto-Sivashinsky equation, J. Dynam. Diff. Eqs. (1990), in press · Zbl 0704.58030 |
| [16] | X. Liu, On approximate inertial manifolds for two and three dimensional turbulent flows, J. Math. Anal. Appl., to appear.; X. Liu, On approximate inertial manifolds for two and three dimensional turbulent flows, J. Math. Anal. Appl., to appear. · Zbl 0809.35075 |
| [17] | Nicolaenko, B.; Scheurer, B.; Temam, R., Some global dynamical properties of the Kuramoto-Sivashinsky equation: Nonlinear stability and attractors, Physica D, 16, 155-183 (1985) · Zbl 0592.35013 |
| [18] | Temam, R., Infinite Dimensional Dynamics Systems in Mechanics and Physics (1988), Springer: Springer Berlin · Zbl 0662.35001 |
| [19] | Titi, E. S., Une variété approximante de l’attracteau universel des équations de Navier-Stokes, non linéaire, de dimension finie, Compt. Rend. Acad. Sci. Paris I, 307, 383-385 (1988) · Zbl 0683.35073 |
| [20] | E.S. Titi, On approximate inertial manifolds to the Navier-Stokes equations, MSI Preprint 88-119, J. Math. Anal. Appl., to appear.; E.S. Titi, On approximate inertial manifolds to the Navier-Stokes equations, MSI Preprint 88-119, J. Math. Anal. Appl., to appear. |
| [21] | E.S. Titi, private communication.; E.S. Titi, private communication. |
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